Mathematics for the interested outsider

The Exponential Series

What is it that makes the exponential what it is? We defined it as the inverse of the logarithm, and thisis defined by integrating . But the important thing we immediately showed is that it satisfies the exponential property.

In fact, we can work out the series around any other point the same way. Since all the derivatives are the exponential function back again, we find

Or we could also write this by expanding around and writing the relation as a series in the displacement :

Then we can expand out the part as a series itself:

But then (with our usual handwaving about rearranging series) we can pull out the inner series since it doesn’t depend on the outer summation variable at all:

And these series are just the series defining and , respectively. That is, we have shown the exponential property directly from the series expansion.

That is, whatever function the power series defines, it satisfies the exponential property. In a sense, the fact that the inverse of this function turns out to be the logarithm is a big coincidence. But it’s a coincidence we’ll tease out tomorrow.

For now I’ll note that this important exponential property follows directly from the series. And we can write down the series anywhere we can add, subtract, multiply, divide (at least by integers), and talk about convergence. That is, the exponential series makes sense in any topological ring of characteristic zero. For example, we can define the exponential of complex numbers by the series

Finally, this series will have the exponential property as above, so long as the ring is commutative (like it is for the complex numbers). In more general rings there’s a generalized version of the exponential property, but I’ll leave that until we eventually need to use it.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.